Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 11.7s

Alternatives: 12

Speedup: 4.0×

• 99.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{x}}{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (sqrt (/ (pow (pow (exp x) 2.0) x) PI))
  (+
   (* 0.75 (pow x -5.0))
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (pow x 2.0))) x)))))

C

double code(double x) {
	return sqrt((pow(pow(exp(x), 2.0), x) / ((double) M_PI))) * ((0.75 * pow(x, -5.0)) + fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / pow(x, 2.0))) / x)));
}

Julia

function code(x)
	return Float64(sqrt(Float64(((exp(x) ^ 2.0) ^ x) / pi)) * Float64(Float64(0.75 * (x ^ -5.0)) + fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / (x ^ 2.0))) / x))))
end

Wolfram

code[x_] := N[(N[Sqrt[N[(N[Power[N[Power[N[Exp[x], $MachinePrecision], 2.0], $MachinePrecision], x], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.75 * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]

   2. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{\color{blue}{-5}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. fabs-sqr 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   12. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)} \]

7. Taylor expanded in x around inf 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \color{blue}{\frac{1 + 0.5 \cdot \frac{1}{{x}^{2}}}{x}}\right)\right) \]
8. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{x}\right)\right) \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\color{blue}{0.5}}{{x}^{2}}}{x}\right)\right) \]

9. Simplified 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \color{blue}{\frac{1 + \frac{0.5}{{x}^{2}}}{x}}\right)\right) \]
10. Step-by-step derivation

    1. pow-exp 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    2. sqr-pow 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    3. pow-prod-down 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x} \cdot e^{x}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    4. pow2 100.0%

       \[\leadsto \frac{{\color{blue}{\left({\left(e^{x}\right)}^{2}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]
12. Step-by-step derivation

    1. *-un-lft-identity 100.0%

       \[\leadsto \color{blue}{\left(1 \cdot \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}}\right)} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    2. add-sqr-sqrt 100.0%

       \[\leadsto \left(1 \cdot \color{blue}{\left(\sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}}} \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}}}\right)}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    3. sqrt-unprod 100.0%

       \[\leadsto \left(1 \cdot \color{blue}{\sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    4. frac-times 100.0%

       \[\leadsto \left(1 \cdot \sqrt{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi} \cdot \sqrt{\pi}}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    5. pow-prod-up 100.0%

       \[\leadsto \left(1 \cdot \sqrt{\frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2} + \frac{x}{2}\right)}}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    6. add-log-exp 100.0%

       \[\leadsto \left(1 \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\color{blue}{\log \left(e^{\frac{x}{2}}\right)} + \frac{x}{2}\right)}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    7. exp-sqrt 100.0%

       \[\leadsto \left(1 \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\log \color{blue}{\left(\sqrt{e^{x}}\right)} + \frac{x}{2}\right)}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    8. add-log-exp 100.0%

       \[\leadsto \left(1 \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\log \left(\sqrt{e^{x}}\right) + \color{blue}{\log \left(e^{\frac{x}{2}}\right)}\right)}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    9. exp-sqrt 100.0%

       \[\leadsto \left(1 \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\log \left(\sqrt{e^{x}}\right) + \log \color{blue}{\left(\sqrt{e^{x}}\right)}\right)}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    10. log-prod 100.0%

        \[\leadsto \left(1 \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\color{blue}{\log \left(\sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)}}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    11. add-sqr-sqrt 100.0%

        \[\leadsto \left(1 \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\log \color{blue}{\left(e^{x}\right)}}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    12. add-log-exp 100.0%

        \[\leadsto \left(1 \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\color{blue}{x}}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    13. add-sqr-sqrt 100.0%

        \[\leadsto \left(1 \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{x}}{\color{blue}{\pi}}}\right) \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

13. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{x}}{\pi}}\right)} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]
14. Step-by-step derivation

    1. *-lft-identity 100.0%

       \[\leadsto \color{blue}{\sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{x}}{\pi}}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

15. Simplified 100.0%

    \[\leadsto \color{blue}{\sqrt{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{x}}{\pi}}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]
16. Add Preprocessing

Alternative 2: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \cdot \frac{{\left(e^{x \cdot 2}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   (* 0.75 (pow x -5.0))
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (pow x 2.0))) x)))
  (/ (pow (exp (* x 2.0)) (/ x 2.0)) (sqrt PI))))

C

double code(double x) {
	return ((0.75 * pow(x, -5.0)) + fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / pow(x, 2.0))) / x))) * (pow(exp((x * 2.0)), (x / 2.0)) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(Float64(Float64(0.75 * (x ^ -5.0)) + fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / (x ^ 2.0))) / x))) * Float64((exp(Float64(x * 2.0)) ^ Float64(x / 2.0)) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(N[(0.75 * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[N[(x * 2.0), $MachinePrecision]], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]

   2. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{\color{blue}{-5}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. fabs-sqr 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   12. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)} \]

7. Taylor expanded in x around inf 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \color{blue}{\frac{1 + 0.5 \cdot \frac{1}{{x}^{2}}}{x}}\right)\right) \]
8. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{x}\right)\right) \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\color{blue}{0.5}}{{x}^{2}}}{x}\right)\right) \]

9. Simplified 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \color{blue}{\frac{1 + \frac{0.5}{{x}^{2}}}{x}}\right)\right) \]
10. Step-by-step derivation

    1. pow-exp 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    2. sqr-pow 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    3. pow-prod-down 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x} \cdot e^{x}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

    4. pow2 100.0%

       \[\leadsto \frac{{\color{blue}{\left({\left(e^{x}\right)}^{2}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]
12. Step-by-step derivation

    1. pow-exp 100.0%

       \[\leadsto \frac{{\color{blue}{\left(e^{x \cdot 2}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

13. Applied egg-rr 100.0%

    \[\leadsto \frac{{\color{blue}{\left(e^{x \cdot 2}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

14. Final simplification 100.0%

    \[\leadsto \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \cdot \frac{{\left(e^{x \cdot 2}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \]
15. Add Preprocessing

Alternative 3: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   (* 0.75 (pow x -5.0))
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (pow x 2.0))) x)))
  (/ (pow (exp x) x) (sqrt PI))))

C

double code(double x) {
	return ((0.75 * pow(x, -5.0)) + fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / pow(x, 2.0))) / x))) * (pow(exp(x), x) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(Float64(Float64(0.75 * (x ^ -5.0)) + fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / (x ^ 2.0))) / x))) * Float64((exp(x) ^ x) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(N[(0.75 * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]

   2. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{\color{blue}{-5}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. fabs-sqr 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   12. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)} \]

7. Taylor expanded in x around inf 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \color{blue}{\frac{1 + 0.5 \cdot \frac{1}{{x}^{2}}}{x}}\right)\right) \]
8. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{x}\right)\right) \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\color{blue}{0.5}}{{x}^{2}}}{x}\right)\right) \]

9. Simplified 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \color{blue}{\frac{1 + \frac{0.5}{{x}^{2}}}{x}}\right)\right) \]
10. Step-by-step derivation

    1. pow-exp 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \]

12. Final simplification 100.0%

    \[\leadsto \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{{x}^{2}}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 4: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+
   (* 0.5 (pow x -3.0))
   (/ (+ 1.0 (fma 0.75 (pow x -4.0) (* 1.875 (pow x -6.0)))) x))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * ((0.5 * pow(x, -3.0)) + ((1.0 + fma(0.75, pow(x, -4.0), (1.875 * pow(x, -6.0)))) / x));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(0.5 * (x ^ -3.0)) + Float64(Float64(1.0 + fma(0.75, (x ^ -4.0), Float64(1.875 * (x ^ -6.0)))) / x)))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.75 * N[Power[x, -4.0], $MachinePrecision] + N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]

   2. pow-flip 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-3\right)}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-3\right)} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   4. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-3\right)} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {\color{blue}{x}}^{\left(-3\right)} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{\color{blue}{-3}} + \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \]

   7. associate-*l/ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-3} + \color{blue}{\frac{1 \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)}{\left|x\right|}}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)} \]
7. Add Preprocessing

Alternative 5: 100.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (+
   (* 0.75 (pow x -5.0))
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (* x x))) x)))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * ((0.75 * pow(x, -5.0)) + fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / (x * x))) / x)));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(0.75 * (x ^ -5.0)) + fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / x))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.75 * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]

   2. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{\color{blue}{-5}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. fabs-sqr 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   12. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)} \]

7. Taylor expanded in x around inf 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \color{blue}{\frac{1 + 0.5 \cdot \frac{1}{{x}^{2}}}{x}}\right)\right) \]
8. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{x}\right)\right) \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{\color{blue}{0.5}}{{x}^{2}}}{x}\right)\right) \]

9. Simplified 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \color{blue}{\frac{1 + \frac{0.5}{{x}^{2}}}{x}}\right)\right) \]
10. Step-by-step derivation

    1. unpow2 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{\color{blue}{x \cdot x}}}{x}\right)\right) \]

11. Applied egg-rr 99.9%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{\color{blue}{x \cdot x}}}{x}\right)\right) \]
12. Add Preprocessing

Alternative 6: 99.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (+ (* 0.75 (pow x -5.0)) (fma 1.875 (pow x -7.0) (/ 1.0 x)))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * ((0.75 * pow(x, -5.0)) + fma(1.875, pow(x, -7.0), (1.0 / x)));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(0.75 * (x ^ -5.0)) + fma(1.875, (x ^ -7.0), Float64(1.0 / x))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.75 * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]

   2. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{\color{blue}{-5}} + \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. inv-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. pow-pow 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. fabs-sqr 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   12. add-sqr-sqrt 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)} \]

7. Taylor expanded in x around inf 98.6%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(0.75 \cdot {x}^{-5} + \mathsf{fma}\left(1.875, {x}^{-7}, \color{blue}{\frac{1}{x}}\right)\right) \]
8. Add Preprocessing

Alternative 7: 52.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{\frac{0.5}{x}}{x}}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (/ (exp (pow x 2.0)) (sqrt PI)) (/ (/ 0.5 x) x)) (fabs x)))

C

double code(double x) {
	return ((exp(pow(x, 2.0)) / sqrt(((double) M_PI))) * ((0.5 / x) / x)) / fabs(x);
}

Java

public static double code(double x) {
	return ((Math.exp(Math.pow(x, 2.0)) / Math.sqrt(Math.PI)) * ((0.5 / x) / x)) / Math.abs(x);
}

Python

def code(x):
	return ((math.exp(math.pow(x, 2.0)) / math.sqrt(math.pi)) * ((0.5 / x) / x)) / math.fabs(x)

Julia

function code(x)
	return Float64(Float64(Float64(exp((x ^ 2.0)) / sqrt(pi)) * Float64(Float64(0.5 / x) / x)) / abs(x))
end

MATLAB

function tmp = code(x)
	tmp = ((exp((x ^ 2.0)) / sqrt(pi)) * ((0.5 / x) / x)) / abs(x);
end

Wolfram

code[x_] := N[(N[(N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 34.3%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{0.5}{{x}^{2} \cdot \left|x\right|}} \]
6. Step-by-step derivation

   1. associate-/r* 36.2%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{{x}^{2}}}{\left|x\right|}} \]

7. Simplified 36.2%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{{x}^{2}}}{\left|x\right|}} \]
8. Step-by-step derivation

   1. associate-*r/ 49.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{0.5}{{x}^{2}}}{\left|x\right|}} \]

   2. pow2 49.9%

      \[\leadsto \frac{\frac{e^{\color{blue}{{x}^{2}}}}{\sqrt{\pi}} \cdot \frac{0.5}{{x}^{2}}}{\left|x\right|} \]

   3. div-inv 49.9%

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}}\right)}}{\left|x\right|} \]

   4. pow-flip 51.9%

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot \color{blue}{{x}^{\left(-2\right)}}\right)}{\left|x\right|} \]

   5. metadata-eval 51.9%

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{\color{blue}{-2}}\right)}{\left|x\right|} \]

9. Applied egg-rr 51.9%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-2}\right)}{\left|x\right|}} \]
10. Step-by-step derivation

    1. metadata-eval 51.9%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{\color{blue}{\left(-2\right)}}\right)}{\left|x\right|} \]

    2. pow-flip 49.9%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)}{\left|x\right|} \]

    3. div-inv 49.9%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\frac{0.5}{{x}^{2}}}}{\left|x\right|} \]

    4. pow2 49.9%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5}{\color{blue}{x \cdot x}}}{\left|x\right|} \]

    5. associate-/r* 52.3%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{x}}{x}}}{\left|x\right|} \]

11. Applied egg-rr 52.3%

    \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{x}}{x}}}{\left|x\right|} \]
12. Add Preprocessing

Alternative 8: 34.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot \frac{{x}^{-2}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (exp (pow x 2.0)) (sqrt PI)) (* 0.5 (/ (pow x -2.0) x))))

C

double code(double x) {
	return (exp(pow(x, 2.0)) / sqrt(((double) M_PI))) * (0.5 * (pow(x, -2.0) / x));
}

Java

public static double code(double x) {
	return (Math.exp(Math.pow(x, 2.0)) / Math.sqrt(Math.PI)) * (0.5 * (Math.pow(x, -2.0) / x));
}

Python

def code(x):
	return (math.exp(math.pow(x, 2.0)) / math.sqrt(math.pi)) * (0.5 * (math.pow(x, -2.0) / x))

Julia

function code(x)
	return Float64(Float64(exp((x ^ 2.0)) / sqrt(pi)) * Float64(0.5 * Float64((x ^ -2.0) / x)))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x ^ 2.0)) / sqrt(pi)) * (0.5 * ((x ^ -2.0) / x));
end

Wolfram

code[x_] := N[(N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[Power[x, -2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 34.3%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{0.5}{{x}^{2} \cdot \left|x\right|}} \]
6. Step-by-step derivation

   1. associate-/r* 36.2%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{{x}^{2}}}{\left|x\right|}} \]

7. Simplified 36.2%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{{x}^{2}}}{\left|x\right|}} \]
8. Step-by-step derivation

   1. associate-*r/ 49.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{0.5}{{x}^{2}}}{\left|x\right|}} \]

   2. pow2 49.9%

      \[\leadsto \frac{\frac{e^{\color{blue}{{x}^{2}}}}{\sqrt{\pi}} \cdot \frac{0.5}{{x}^{2}}}{\left|x\right|} \]

   3. div-inv 49.9%

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}}\right)}}{\left|x\right|} \]

   4. pow-flip 51.9%

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot \color{blue}{{x}^{\left(-2\right)}}\right)}{\left|x\right|} \]

   5. metadata-eval 51.9%

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{\color{blue}{-2}}\right)}{\left|x\right|} \]

9. Applied egg-rr 51.9%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-2}\right)}{\left|x\right|}} \]
10. Step-by-step derivation

    1. div-inv 51.9%

       \[\leadsto \color{blue}{\left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-2}\right)\right) \cdot \frac{1}{\left|x\right|}} \]

    2. add-sqr-sqrt 51.9%

       \[\leadsto \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-2}\right)\right) \cdot \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \]

    3. fabs-sqr 51.9%

       \[\leadsto \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-2}\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

    4. add-sqr-sqrt 51.9%

       \[\leadsto \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-2}\right)\right) \cdot \frac{1}{\color{blue}{x}} \]

11. Applied egg-rr 51.9%

    \[\leadsto \color{blue}{\left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-2}\right)\right) \cdot \frac{1}{x}} \]
12. Step-by-step derivation

    1. associate-*l* 36.2%

       \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(\left(0.5 \cdot {x}^{-2}\right) \cdot \frac{1}{x}\right)} \]

    2. associate-*r/ 36.2%

       \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\left(0.5 \cdot {x}^{-2}\right) \cdot 1}{x}} \]

    3. *-rgt-identity 36.2%

       \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{\color{blue}{0.5 \cdot {x}^{-2}}}{x} \]

    4. associate-*r/ 36.2%

       \[\leadsto \frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{{x}^{-2}}{x}\right)} \]

13. Simplified 36.2%

    \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot \frac{{x}^{-2}}{x}\right)} \]
14. Add Preprocessing

Alternative 9: 33.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{{x}^{2}} \cdot \frac{0.5}{{x}^{3}}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (exp (pow x 2.0)) (/ 0.5 (pow x 3.0))) (sqrt PI)))

C

double code(double x) {
	return (exp(pow(x, 2.0)) * (0.5 / pow(x, 3.0))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (Math.exp(Math.pow(x, 2.0)) * (0.5 / Math.pow(x, 3.0))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (math.exp(math.pow(x, 2.0)) * (0.5 / math.pow(x, 3.0))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(exp((x ^ 2.0)) * Float64(0.5 / (x ^ 3.0))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x ^ 2.0)) * (0.5 / (x ^ 3.0))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] * N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 34.3%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{0.5}{{x}^{2} \cdot \left|x\right|}} \]
6. Step-by-step derivation

   1. associate-/r* 36.2%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{{x}^{2}}}{\left|x\right|}} \]

7. Simplified 36.2%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{{x}^{2}}}{\left|x\right|}} \]
8. Step-by-step derivation

   1. associate-*r/ 49.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{0.5}{{x}^{2}}}{\left|x\right|}} \]

   2. pow2 49.9%

      \[\leadsto \frac{\frac{e^{\color{blue}{{x}^{2}}}}{\sqrt{\pi}} \cdot \frac{0.5}{{x}^{2}}}{\left|x\right|} \]

   3. div-inv 49.9%

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}}\right)}}{\left|x\right|} \]

   4. pow-flip 51.9%

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot \color{blue}{{x}^{\left(-2\right)}}\right)}{\left|x\right|} \]

   5. metadata-eval 51.9%

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{\color{blue}{-2}}\right)}{\left|x\right|} \]

9. Applied egg-rr 51.9%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-2}\right)}{\left|x\right|}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 51.9%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(0.5 \cdot {x}^{-2}\right)}{\left|x\right|}} \]

    2. associate-/l* 36.2%

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5 \cdot {x}^{-2}}{\left|x\right|}\right)} \]

    3. add-sqr-sqrt 36.2%

       \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5 \cdot {x}^{-2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right) \]

    4. fabs-sqr 36.2%

       \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5 \cdot {x}^{-2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right) \]

    5. add-sqr-sqrt 36.2%

       \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5 \cdot {x}^{-2}}{\color{blue}{x}}\right) \]

    6. metadata-eval 36.2%

       \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5 \cdot {x}^{\color{blue}{\left(-2\right)}}}{x}\right) \]

    7. pow-flip 36.2%

       \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5 \cdot \color{blue}{\frac{1}{{x}^{2}}}}{x}\right) \]

    8. div-inv 36.2%

       \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{\color{blue}{\frac{0.5}{{x}^{2}}}}{x}\right) \]

    9. add-sqr-sqrt 36.2%

       \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right) \]

    10. fabs-sqr 36.2%

        \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}}\right) \]

    11. add-sqr-sqrt 36.2%

        \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{\frac{0.5}{{x}^{2}}}{\left|\color{blue}{x}\right|}\right) \]

    12. associate-/l/ 34.3%

        \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \color{blue}{\frac{0.5}{\left|x\right| \cdot {x}^{2}}}\right) \]

    13. add-sqr-sqrt 34.3%

        \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {x}^{2}}\right) \]

    14. fabs-sqr 34.3%

        \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5}{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {x}^{2}}\right) \]

    15. add-sqr-sqrt 34.3%

        \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5}{\color{blue}{x} \cdot {x}^{2}}\right) \]

    16. pow2 34.3%

        \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right) \]

    17. cube-mult 34.3%

        \[\leadsto 1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5}{\color{blue}{{x}^{3}}}\right) \]

11. Applied egg-rr 34.3%

    \[\leadsto \color{blue}{1 \cdot \left(\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5}{{x}^{3}}\right)} \]
12. Step-by-step derivation

    1. *-lft-identity 34.3%

       \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \frac{0.5}{{x}^{3}}} \]

    2. *-commutative 34.3%

       \[\leadsto \color{blue}{\frac{0.5}{{x}^{3}} \cdot \frac{e^{{x}^{2}}}{\sqrt{\pi}}} \]

    3. associate-*r/ 34.3%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{{x}^{3}} \cdot e^{{x}^{2}}}{\sqrt{\pi}}} \]

13. Simplified 34.3%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{{x}^{3}} \cdot e^{{x}^{2}}}{\sqrt{\pi}}} \]

14. Final simplification 34.3%

    \[\leadsto \frac{e^{{x}^{2}} \cdot \frac{0.5}{{x}^{3}}}{\sqrt{\pi}} \]
15. Add Preprocessing

Alternative 10: 14.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{1.875}{{x}^{7}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (pow (exp x) x) (sqrt PI)) (/ 1.875 (pow x 7.0))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * (1.875 / pow(x, 7.0));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * (1.875 / Math.pow(x, 7.0));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * (1.875 / math.pow(x, 7.0))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(1.875 / (x ^ 7.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * (1.875 / (x ^ 7.0));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}\right) \]

   2. expm1-undefine 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)} - 1\right)}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)\right)} - 1\right)}\right) \]
7. Step-by-step derivation

   1. sub-neg 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)\right)} + \left(-1\right)\right)}\right) \]

   2. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(e^{\mathsf{log1p}\left(1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)\right)} + \color{blue}{-1}\right)\right) \]

   3. +-commutative 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)\right)}\right)}\right) \]

   4. log1p-undefine 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(-1 + e^{\color{blue}{\log \left(1 + \left(1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)\right)\right)}}\right)\right) \]

   5. rem-exp-log 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(-1 + \color{blue}{\left(1 + \left(1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)\right)\right)}\right)\right) \]

   6. associate-+r+ 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(-1 + \color{blue}{\left(\left(1 + 1\right) + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)\right)}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(-1 + \left(\color{blue}{2} + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)\right)\right)\right) \]

8. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \color{blue}{\left(-1 + \left(2 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)\right)\right)}\right) \]

9. Taylor expanded in x around 0 15.9%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1.875}{{x}^{6} \cdot \left|x\right|}} \]
10. Step-by-step derivation

    1. associate-/r* 15.9%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{1.875}{{x}^{6}}}{\left|x\right|}} \]

    2. add-sqr-sqrt 15.9%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{\frac{1.875}{{x}^{6}}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \]

    3. fabs-sqr 15.9%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{\frac{1.875}{{x}^{6}}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

    4. add-sqr-sqrt 15.9%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{\frac{1.875}{{x}^{6}}}{\color{blue}{x}} \]

    5. *-un-lft-identity 15.9%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(1 \cdot \frac{\frac{1.875}{{x}^{6}}}{x}\right)} \]

11. Applied egg-rr 15.9%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\left(1 \cdot \frac{\frac{1.875}{{x}^{6}}}{x}\right)} \]
12. Step-by-step derivation

    1. *-lft-identity 15.9%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{1.875}{{x}^{6}}}{x}} \]

    2. associate-/r* 15.9%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1.875}{{x}^{6} \cdot x}} \]

    3. pow-plus 15.9%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{1.875}{\color{blue}{{x}^{\left(6 + 1\right)}}} \]

    4. metadata-eval 15.9%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{1.875}{{x}^{\color{blue}{7}}} \]

13. Simplified 15.9%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1.875}{{x}^{7}}} \]
14. Add Preprocessing

Alternative 11: 1.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{{x}^{-3}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (/ (pow x -3.0) (sqrt PI))))

C

double code(double x) {
	return 0.5 * (pow(x, -3.0) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return 0.5 * (Math.pow(x, -3.0) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return 0.5 * (math.pow(x, -3.0) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(0.5 * Float64((x ^ -3.0) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x ^ -3.0) / sqrt(pi));
end

Wolfram

code[x_] := N[(0.5 * N[(N[Power[x, -3.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 2.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
6. Step-by-step derivation

   1. associate-*l/ 2.0%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{\pi}}}{{x}^{2} \cdot \left|x\right|}} \]

   2. *-lft-identity 2.0%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\frac{1}{\pi}}}}{{x}^{2} \cdot \left|x\right|} \]

   3. associate-*r/ 2.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{{x}^{2} \cdot \left|x\right|}} \]

   4. unpow2 2.0%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|} \]

   5. sqr-abs 2.0%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left|x\right|} \]

   6. unpow3 2.0%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{\color{blue}{{\left(\left|x\right|\right)}^{3}}} \]

7. Simplified 2.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{{\left(\left|x\right|\right)}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 2.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}} \]

   2. inv-pow 2.0%

      \[\leadsto \left(0.5 \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right) \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} \]

   3. sqrt-pow1 2.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right) \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} \]

   4. metadata-eval 2.0%

      \[\leadsto \left(0.5 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} \]

   5. pow-flip 2.0%

      \[\leadsto \left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-3\right)}} \]

   6. metadata-eval 2.0%

      \[\leadsto \left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{\color{blue}{-3}} \]

9. Applied egg-rr 2.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 2.0%

       \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}} \cdot \sqrt{\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}}} \]

    2. sqrt-unprod 2.0%

       \[\leadsto \color{blue}{\sqrt{\left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}\right) \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}\right)}} \]

    3. *-commutative 2.0%

       \[\leadsto \sqrt{\color{blue}{\left({\left(\left|x\right|\right)}^{-3} \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}\right)} \]

    4. *-commutative 2.0%

       \[\leadsto \sqrt{\left({\left(\left|x\right|\right)}^{-3} \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right) \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{-3} \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)}} \]

    5. swap-sqr 2.0%

       \[\leadsto \sqrt{\color{blue}{\left({\left(\left|x\right|\right)}^{-3} \cdot {\left(\left|x\right|\right)}^{-3}\right) \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)}} \]

    6. pow-prod-down 2.0%

       \[\leadsto \sqrt{\color{blue}{{\left(\left|x\right| \cdot \left|x\right|\right)}^{-3}} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    7. sqr-abs 2.0%

       \[\leadsto \sqrt{{\color{blue}{\left(x \cdot x\right)}}^{-3} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    8. pow-prod-down 2.0%

       \[\leadsto \sqrt{\color{blue}{\left({x}^{-3} \cdot {x}^{-3}\right)} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    9. pow-sqr 2.0%

       \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 \cdot -3\right)}} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    10. metadata-eval 2.0%

        \[\leadsto \sqrt{{x}^{\color{blue}{-6}} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    11. *-commutative 2.0%

        \[\leadsto \sqrt{{x}^{-6} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot 0.5\right)}\right)} \]

    12. metadata-eval 2.0%

        \[\leadsto \sqrt{{x}^{-6} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.5\right)\right)} \]

    13. sqrt-pow1 2.0%

        \[\leadsto \sqrt{{x}^{-6} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot 0.5\right)\right)} \]

    14. inv-pow 2.0%

        \[\leadsto \sqrt{{x}^{-6} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot 0.5\right)\right)} \]

11. Applied egg-rr 2.0%

    \[\leadsto \color{blue}{\sqrt{{x}^{-6} \cdot \left(\frac{1}{\pi} \cdot 0.25\right)}} \]
12. Step-by-step derivation

    1. associate-*l/ 2.0%

       \[\leadsto \sqrt{{x}^{-6} \cdot \color{blue}{\frac{1 \cdot 0.25}{\pi}}} \]

    2. metadata-eval 2.0%

       \[\leadsto \sqrt{{x}^{-6} \cdot \frac{\color{blue}{0.25}}{\pi}} \]

13. Simplified 2.0%

    \[\leadsto \color{blue}{\sqrt{{x}^{-6} \cdot \frac{0.25}{\pi}}} \]
14. Step-by-step derivation

    1. *-un-lft-identity 2.0%

       \[\leadsto \color{blue}{1 \cdot \sqrt{{x}^{-6} \cdot \frac{0.25}{\pi}}} \]

    2. sqrt-prod 2.0%

       \[\leadsto 1 \cdot \color{blue}{\left(\sqrt{{x}^{-6}} \cdot \sqrt{\frac{0.25}{\pi}}\right)} \]

    3. sqrt-pow1 2.0%

       \[\leadsto 1 \cdot \left(\color{blue}{{x}^{\left(\frac{-6}{2}\right)}} \cdot \sqrt{\frac{0.25}{\pi}}\right) \]

    4. metadata-eval 2.0%

       \[\leadsto 1 \cdot \left({x}^{\color{blue}{-3}} \cdot \sqrt{\frac{0.25}{\pi}}\right) \]

    5. sqrt-div 2.0%

       \[\leadsto 1 \cdot \left({x}^{-3} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{\pi}}}\right) \]

    6. metadata-eval 2.0%

       \[\leadsto 1 \cdot \left({x}^{-3} \cdot \frac{\color{blue}{0.5}}{\sqrt{\pi}}\right) \]

15. Applied egg-rr 2.0%

    \[\leadsto \color{blue}{1 \cdot \left({x}^{-3} \cdot \frac{0.5}{\sqrt{\pi}}\right)} \]
16. Step-by-step derivation

    1. *-lft-identity 2.0%

       \[\leadsto \color{blue}{{x}^{-3} \cdot \frac{0.5}{\sqrt{\pi}}} \]

    2. *-commutative 2.0%

       \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\pi}} \cdot {x}^{-3}} \]

    3. associate-*l/ 2.0%

       \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-3}}{\sqrt{\pi}}} \]

    4. associate-/l* 2.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{-3}}{\sqrt{\pi}}} \]

17. Simplified 2.0%

    \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{-3}}{\sqrt{\pi}}} \]
18. Add Preprocessing

Alternative 12: 1.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{{x}^{-6} \cdot \frac{0.25}{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* (pow x -6.0) (/ 0.25 PI))))

C

double code(double x) {
	return sqrt((pow(x, -6.0) * (0.25 / ((double) M_PI))));
}

Java

public static double code(double x) {
	return Math.sqrt((Math.pow(x, -6.0) * (0.25 / Math.PI)));
}

Python

def code(x):
	return math.sqrt((math.pow(x, -6.0) * (0.25 / math.pi)))

Julia

function code(x)
	return sqrt(Float64((x ^ -6.0) * Float64(0.25 / pi)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((x ^ -6.0) * (0.25 / pi)));
end

Wolfram

code[x_] := N[Sqrt[N[(N[Power[x, -6.0], $MachinePrecision] * N[(0.25 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 2.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
6. Step-by-step derivation

   1. associate-*l/ 2.0%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{\pi}}}{{x}^{2} \cdot \left|x\right|}} \]

   2. *-lft-identity 2.0%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\frac{1}{\pi}}}}{{x}^{2} \cdot \left|x\right|} \]

   3. associate-*r/ 2.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{{x}^{2} \cdot \left|x\right|}} \]

   4. unpow2 2.0%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|} \]

   5. sqr-abs 2.0%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left|x\right|} \]

   6. unpow3 2.0%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{\color{blue}{{\left(\left|x\right|\right)}^{3}}} \]

7. Simplified 2.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{\pi}}}{{\left(\left|x\right|\right)}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 2.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}} \]

   2. inv-pow 2.0%

      \[\leadsto \left(0.5 \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right) \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} \]

   3. sqrt-pow1 2.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right) \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} \]

   4. metadata-eval 2.0%

      \[\leadsto \left(0.5 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} \]

   5. pow-flip 2.0%

      \[\leadsto \left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-3\right)}} \]

   6. metadata-eval 2.0%

      \[\leadsto \left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{\color{blue}{-3}} \]

9. Applied egg-rr 2.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 2.0%

       \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}} \cdot \sqrt{\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}}} \]

    2. sqrt-unprod 2.0%

       \[\leadsto \color{blue}{\sqrt{\left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}\right) \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}\right)}} \]

    3. *-commutative 2.0%

       \[\leadsto \sqrt{\color{blue}{\left({\left(\left|x\right|\right)}^{-3} \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot {\left(\left|x\right|\right)}^{-3}\right)} \]

    4. *-commutative 2.0%

       \[\leadsto \sqrt{\left({\left(\left|x\right|\right)}^{-3} \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right) \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{-3} \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)}} \]

    5. swap-sqr 2.0%

       \[\leadsto \sqrt{\color{blue}{\left({\left(\left|x\right|\right)}^{-3} \cdot {\left(\left|x\right|\right)}^{-3}\right) \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)}} \]

    6. pow-prod-down 2.0%

       \[\leadsto \sqrt{\color{blue}{{\left(\left|x\right| \cdot \left|x\right|\right)}^{-3}} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    7. sqr-abs 2.0%

       \[\leadsto \sqrt{{\color{blue}{\left(x \cdot x\right)}}^{-3} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    8. pow-prod-down 2.0%

       \[\leadsto \sqrt{\color{blue}{\left({x}^{-3} \cdot {x}^{-3}\right)} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    9. pow-sqr 2.0%

       \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 \cdot -3\right)}} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    10. metadata-eval 2.0%

        \[\leadsto \sqrt{{x}^{\color{blue}{-6}} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(0.5 \cdot {\pi}^{-0.5}\right)\right)} \]

    11. *-commutative 2.0%

        \[\leadsto \sqrt{{x}^{-6} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot 0.5\right)}\right)} \]

    12. metadata-eval 2.0%

        \[\leadsto \sqrt{{x}^{-6} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.5\right)\right)} \]

    13. sqrt-pow1 2.0%

        \[\leadsto \sqrt{{x}^{-6} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot 0.5\right)\right)} \]

    14. inv-pow 2.0%

        \[\leadsto \sqrt{{x}^{-6} \cdot \left(\left(0.5 \cdot {\pi}^{-0.5}\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot 0.5\right)\right)} \]

11. Applied egg-rr 2.0%

    \[\leadsto \color{blue}{\sqrt{{x}^{-6} \cdot \left(\frac{1}{\pi} \cdot 0.25\right)}} \]
12. Step-by-step derivation

    1. associate-*l/ 2.0%

       \[\leadsto \sqrt{{x}^{-6} \cdot \color{blue}{\frac{1 \cdot 0.25}{\pi}}} \]

    2. metadata-eval 2.0%

       \[\leadsto \sqrt{{x}^{-6} \cdot \frac{\color{blue}{0.25}}{\pi}} \]

13. Simplified 2.0%

    \[\leadsto \color{blue}{\sqrt{{x}^{-6} \cdot \frac{0.25}{\pi}}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))